PREFACE TO THE ENLARGED AND
CORRECTED PRINTING
This book is the first volume of a treatise which will eventually consist of
four volumes. It is also an enlarged and corrected printing, essentially
without changes, of my " Foundations of Modern Analysis," published in
1960. Many readers, colleagues, and friends have urged me to write a sequel
to that book, and in the end I became convinced that there was a place for
a survey of modern analysis, somewhere between the ''minimum tool kit"
of an elementary nature which I had intended to write, and specialist
monographs leading to the frontiers of research. My experience of teaching
has also persuaded me that the mathematical apprentice, after taking the first
step of " Foundations," needs further guidance and a kind of general bird's
eye-view of his subject before he is launched onto the ocean of mathematical
literature or set on the narrow path of his own topic of research.
Thus I have finally been led to attempt to write an equivalent, for the
mathematicians of 1970, of what the "Cours d'Analyse" of Jordan, Picard,
and Goursat were for mathematical students between 1880 and 1920.
It is manifestly out of the question to attempt encyclopedic coverage, and
certainly superfluous to rewrite the works of N. Bourbaki. I have therefore
been obliged to cut ruthlessly in order to keep within limits comparable to
those of the classical treatises. I have opted for breadth rather than depth, in
the opinion that it is better to show the reader rudiments of many branches
of modern analysis rather than to provide him with a complete and detailed
exposition of a small number of topics.
Experience seems to show that the student usually finds a new theory
difficult to grasp at a first reading. He needs to return to it several times before
he becomes really familiar with it and can distinguish for himself which
are the essential ideas and which results are of minor importance, and only
then will he be able to apply it intelligently. The chapters of this treatise are